Let’s solve each problem one by one, step by step.
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Problem 1:
Hillary swims every 6 days, runs every 4 days, and cycles every 16 days. She did all three today. When will she do all three again on the same day?
We need to find the
Least Common Multiple (LCM) of 6, 4, and 16 — because LCM tells us when all three events line up again.
Step 1: Prime factorization
- 6 = 2 × 3
- 4 = 2²
- 16 = 2⁴
Step 2: Take the highest power of each prime:
- For 2 → highest is 2⁴ (from 16)
- For 3 → highest is 3¹ (from 6)
So LCM = 2⁴ × 3 = 16 × 3 =
48
✔ Answer: In
48 days
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Problem 2:
Oscar has 14 rock CDs, 12 classical CDs, and 8 pop CDs. He wants to pack them so each box has only one type of CD, and each box has the same number of CDs. What’s the greatest number he can put in each box?
This asks for the
Greatest Common Factor (GCF) of 14, 12, and 8 — because we want the largest number that divides all three evenly.
Step 1: List factors or use prime factorization.
Prime factors:
- 14 = 2 × 7
- 12 = 2² × 3
- 8 = 2³
Common prime factor? Only
2, and the lowest power is 2¹.
So GCF =
2
✔ Answer:
2 CDs per box
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Problem 3:
You have 45 sunflower plants, 81 corn plants, and 63 tomato plants. You want to plant them in rows with the same number of plants per row, and each row has only one type of plant. What’s the greatest number of plants you can put in one row?
Again, this is asking for the
GCF of 45, 81, and 63.
Prime factorization:
- 45 = 3² × 5
- 81 = 3⁴
- 63 = 3² × 7
Common prime factor: 3² = 9
✔ Answer:
9 plants per row
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Problem 4:
Cups come in packages of 6, plates in packages of 8. You want the same number of cups and plates. What’s the least number of packages of each you need to buy?
We need the
LCM of 6 and 8 — that gives the smallest number where both quantities match.
Prime factors:
- 6 = 2 × 3
- 8 = 2³
LCM = 2³ × 3 = 8 × 3 =
24
Now, how many packages?
- Cups: 24 ÷ 6 =
4 packages
- Plates: 24 ÷ 8 =
3 packages
✔ Answer:
4 packages of cups and 3 packages of plates
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Problem 5:
A full moon occurs every 30 days. Last full moon was on a Friday. How many days until the next full moon also falls on a Friday?
We need to find when the cycle of “full moon” (every 30 days) lines up with the cycle of “days of the week” (every 7 days).
So we need the
LCM of 30 and 7.
Since 30 and 7 have no common factors (7 is prime, doesn’t divide 30), LCM = 30 × 7 =
210
✔ Answer:
210 days
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Final Answer:
1. 48 days
2. 2 CDs
3. 9 plants
4. 4 packages of cups and 3 packages of plates
5. 210 days
Parent Tip: Review the logic above to help your child master the concept of lcm word problems worksheet.
